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          二级运放设计
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<img src="https://pic.zhouyuqian.com/img/20210727183531.png" alt="电路结构"><figcaption aria-hidden="true">电路结构</figcaption>
</figure>
<p>最基本的 CMOS 二级密勒补偿运算跨导放大器结构如图所示，主要包括四个部分：第一级输入级放大电路、第二级放大电路、偏置电路和相位补偿电路。</p>
<a id="more"></a>
<h1 id="电路分析">电路分析</h1>
<p>输入级电路由 M1~M5 组成；</p>
<p>输出级电路由 M6、M7 组成，M6 为共源放大器，M7 为其提供恒定的偏置电流，同时作为第二级输出负载。相位补偿电路由 M14 和 Cc 构成，M14 工作在线性区，可等效为一个电阻，与电容 Cc 一起跨接在第一级和第二级输出之间，构成 RC 密勒补偿。</p>
<h2 id="静态特性">静态特性</h2>
<p>不考虑 M14，电路的等效模型如下图所示：</p>
<p><img src="https://pic.zhouyuqian.com/img/20210727183651.png" alt="image-20210319101141088" style="zoom:50%;"></p>
<p>第一级和第二级都是跨导放大器，M1 和 M2 相同，则第一级的跨导为： <span class="math display">\[
G_{m1} = g_{m1}=g_{m2}
\]</span></p>
<p>第一级的输出电阻为： <span class="math display">\[
R_1 = r_{o2} || r_{o4}
\]</span> 则第一级的增益为： <span class="math display">\[
A_1 = G_{m1}R_1 = g_{m2}(r_{o2} || r_{o4})
\]</span> 对于第二级，M6、M7 为共源放大器，则： <span class="math display">\[
G_{m2} = g_{m6}=\frac{2I_{DS6}}{V_{GST6}}
\]</span></p>
<p><span class="math display">\[
R_2 = r_{o6}||r_{o7}
\]</span></p>
<p>其中 <span class="math inline">\(V_{GST6} = V_{GS6} - V_{TH6}\)</span>。</p>
<p>则第二级的电压增益为： <span class="math display">\[
A_2 = -G_{m2}R_2 = -g_{m6}(r_{o6}||r_{o7})
\]</span> 故总的直流开环电压增益为： <span class="math display">\[
A_0 = A_1A_2=-g_{m2}g_{m6}(r_{o2} || r_{o4})(r_{o6}||r_{o7})
\]</span> <strong>为了得到较高的增益，应当选取较小的过驱动电压和较大的沟道长度。</strong></p>
<h2 id="频率特性">频率特性</h2>
<p>在等效电路图中，第一级的输出节点的电容 <span class="math inline">\(C_1\)</span> 为： <span class="math display">\[
C_1 = C_{GD2} + C_{DB2} + C_{GD4} + C_{DB4} + C_{GS6}
\]</span> 第二级的输出节点的电容 <span class="math inline">\(C_2\)</span> 为： <span class="math display">\[
C_2 = C_{DB6} + C_{DB7} + C_{GD7} + C_{L}
\]</span> 一般，<span class="math inline">\(C_L\)</span> 远大于晶体管电容，所以 <span class="math inline">\(C_2\)</span> 远大于 <span class="math inline">\(C_1\)</span>，对节点 3 和节点 5 运用 KCL，得： <span class="math display">\[
G_{m1}V_{id}+\frac{V_{i2}}{R_1}+sC_1V_{i2}+sC_c(V_{i2}-V_o) = 0
\]</span></p>
<p><span class="math display">\[
G_{m2}V_{i2}+\frac{V_{o}}{R_2}+sC_2V_{o}+sC_c(V_{o}-V_{12}) = 0
\]</span></p>
<p>联立上面两式，消去中间量 <span class="math inline">\(V_{i2}\)</span> 得到电路的<strong>传输函数</strong>： <span class="math display">\[
\frac{V_o}{V_{id}} = \frac{G_{m1}(G_{m2}-sC_c)R_1R_2}{as^2+bs+1}
\]</span> 其中 <span class="math inline">\(a=[C_1C_2+C_c(C_1+C_2)]R_1R_2 \\ b=C_1R_1+C_2R_2+C_c(G_{m2}R_1R_2+R_1+R_2)\)</span></p>
<p>由传输函数的分子可以得到一个位于<strong>右半平面的零点</strong>： <span class="math display">\[
f_z = \frac{G_{m2}}{2\pi C_c}=\frac{g_{m6}}{2\pi C_c}
\]</span> 分析传输函数的分母，对于 <span class="math inline">\(as^2+bs+c=0\)</span> 的方程，如果有两个相距很远的实根，则 <span class="math inline">\(s_1\approx -c/b, s_2\approx -b/a\)</span>。由此得到： <span class="math display">\[
s_1 = -\frac{1}{C_1R_1+C_2R_2+C_c(G_{m2}R_1R_2+R_1+R_2)}
\]</span></p>
<p><span class="math display">\[
s_2 = -\frac{C_1R_1+C_2R_2+C_c(G_{m2}R_1R_2+R_1+R_2)}{[C_1C_2+C_c(C_1+C_2)]R_1R_2}
\]</span></p>
<p>从而得电路<strong>主极点</strong>为： <span class="math display">\[
f_d = \frac{s_1}{2\pi}=\frac{1}{2\pi R_1 [C_1+C_c(1+G_{m2}R_2)]}
\]</span> 通常 <span class="math inline">\(C_1\)</span> 远小于 <span class="math inline">\(C_c\)</span>，且 <span class="math inline">\(G_{m2}R_2\)</span> 远大于 1，上式可以近似为： <span class="math display">\[
f_d = \frac{1}{2\pi R_1R_2G_{m2}C_c}
\]</span> 而次极点为： <span class="math display">\[
f_{nd} =\frac{s_2}{2\pi}=\frac{G_{m2}C_c}{2\pi (C_1C_2 + C_1C_c+C_2C_c)}
\]</span> 单位增益带宽： <span class="math display">\[
GBW = A_0f_d= G_{m1}G_{m2}R_1R_2 \times \frac{1}{2\pi R_1R_2G_{m2}C_c} = \frac{g_{m2}}{2\pi C_c}
\]</span> 由 <span class="math inline">\(g_m = \mu C_{ox} (W/L) (V_{GS}-VT)\)</span> 得： <span class="math display">\[
GBW = \frac{1}{2\pi C_c}\mu_p C_{ox}(\frac{W}{L})_2V_{GST2}
\]</span> <strong>得到高的 GBW 就需要增大 M1 和 M2 管的过驱动电压或者减小其沟道长度</strong>，<strong>这与提高增益的要求是相互抵触的，而且管子面积的减小也会使得噪声性能变差</strong>，所以在设计电路的时候，需要根据具体应用和设计指标进行权衡。</p>
<p>GBW 也可以写成电流有关的形式 ( <span class="math inline">\(g_m = \mu C_{ox} (W/L) (V_{GS}-VT) = 2I_D/V_{GST}\)</span> )： <span class="math display">\[
GBW = \frac{I_{DS2}}{\pi C_c V_{GST1}}
\]</span> 上式说明，在选定了补偿电容和过驱动电压时，<strong>电路的单位增益带宽与功耗成正比</strong>。而电路设计时一般会对功耗有限制，由此为了得到高的 GBW，需要合理分配各支路的电流。</p>
<p>第二极点可以近似为： <span class="math display">\[
f_{nd} = \frac{g_{m6}}{2\pi C_L}
\]</span> 一般而言，出于电路稳定的需要，<strong>为了达到 45° 的相位裕量，第二极点必须在 1.22GBW 之外，而若要有 60° 相位裕量的话，第二极点必须高于 2.2GWB</strong>。</p>
<h2 id="相位补偿">相位补偿</h2>
<p><span class="math display">\[
\frac{1}{R_zC_c}\approx1.2\frac{g_{m1}}{C_c}
\]</span></p>
<p>即： <span class="math display">\[
g_{m1}R_z\approx\frac{1}{1.2}
\]</span> 若令 <span class="math inline">\(V_{GST1}=V_{GST14}\)</span>，则： <span class="math display">\[
\frac{(W/L)_{14}}{(W/L)_{1}} \approx1.2\frac{\mu _p}{\mu _n}
\]</span></p>
<h2 id="偏置电路">偏置电路</h2>
<p>偏置电路由 M8~M13 构成。</p>
<p>镜像电流源 M8 和 M9 使得 M13 和 M12 的电流相等，都为 <span class="math inline">\(I_B\)</span>，则： <span class="math display">\[
I_B = \frac{1}{2}\mu C_{ox}(\frac{W}{L})_{12}(V_{GS12}-V_T)^2 = \frac{1}{2}\mu C_{ox}(\frac{W}{L})_{13}(V_{GS13}-V_T)^2
\]</span> 而： <span class="math display">\[
V_{GS13} = V_{GS12}+I_BR_B
\]</span> 由上面两式可以得到： <span class="math display">\[
I_B = \frac{2}{\mu C_{ox}(W/L)_{12}R_B^2}(\sqrt{\frac{(W/L)_{12}}{(W/L)_{13}}}-1)^2
\]</span> 可以看到 <span class="math inline">\(I_B\)</span> 仅与电阻 <span class="math inline">\(R_B\)</span> 和 M12、M13 的尺寸有关，不受电源电压的影响。 <span class="math display">\[
R_B = \frac{2}{\sqrt{2\mu C_{ox}(W/L)_{12}I_B}}(\sqrt{\frac{(W/L)_{12}}{(W/L)_{13}}}-1)
\]</span> 由 <span class="math inline">\(\sqrt{2\mu C_{ox}(W/L)_{12}I_B} = g_{m12}\)</span> 以及上式可以得到： <span class="math display">\[
g_{m12}=\frac{2}{R_B}(\sqrt{\frac{(W/L)_{12}}{(W/L)_{13}}}-1)^2
\]</span> 可以看出，<span class="math inline">\(gm_{12}\)</span> 仅由 <span class="math inline">\(R_B\)</span> 以及 M12 与 M13 的器件比例而决定。若取 <span class="math inline">\((W/L)_{12}＝4 (W/L)_{13}\)</span> 则可以得到： <span class="math display">\[
g_{m12} = \frac{2}{R_B}
\]</span></p>
<p><span class="math display">\[
g_{m13} = \frac{1}{R_B}
\]</span></p>
<h1 id="指标分析"><del>指标分析</del></h1>
<h1 id="电路设计">电路设计</h1>
<h2 id="设计指标">设计指标</h2>
<table>
<thead>
<tr class="header">
<th>设计指标</th>
<th>带宽最大化</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>管子长度</td>
<td><span class="math inline">\(\geq 0.8 \mu m\)</span></td>
</tr>
<tr class="even">
<td>管子宽度</td>
<td><span class="math inline">\(\geq 2\mu m\)</span></td>
</tr>
<tr class="odd">
<td>面积</td>
<td><span class="math inline">\(\leq 20000 \mu m^2\)</span></td>
</tr>
<tr class="even">
<td>负载电容</td>
<td><span class="math inline">\(=3pF\)</span></td>
</tr>
<tr class="odd">
<td>共模输入电压</td>
<td>固定在 (VDD+VSS)/2</td>
</tr>
<tr class="even">
<td>输出动态范围</td>
<td>[0.1(VDD-VSS), 0.9(VDD-VSS)]</td>
</tr>
<tr class="odd">
<td>静态功耗</td>
<td><span class="math inline">\(\leq 2mW\)</span></td>
</tr>
<tr class="even">
<td>开环直流增益</td>
<td><span class="math inline">\(\geq 80dB\)</span></td>
</tr>
<tr class="odd">
<td>单位增益带宽</td>
<td>Maximize</td>
</tr>
<tr class="even">
<td>相位裕度</td>
<td><span class="math inline">\(\geq 60 degree\)</span></td>
</tr>
<tr class="odd">
<td>转换速率</td>
<td><span class="math inline">\(\geq 30V/\mu s\)</span></td>
</tr>
<tr class="even">
<td>共模抑制比</td>
<td><span class="math inline">\(\geq 60dB\)</span></td>
</tr>
<tr class="odd">
<td>负电源抑制比</td>
<td><span class="math inline">\(\geq 80dB\)</span></td>
</tr>
<tr class="even">
<td>等效输入噪声</td>
<td><span class="math inline">\(\leq 300nV/Hz@1KHz\)</span></td>
</tr>
</tbody>
</table>
<h2 id="约束">约束</h2>
<p><span class="math display">\[
W_1 = W_2,L_1=L_2;W_3=W_4,L_3=L_4
\]</span></p>
<p><span class="math display">\[
\frac{(W/L)_{3,4}}{(W/L)_{6}} = \frac{\frac{1}{2}(W/L)_{5}}{(W/L)_{7}}
\]</span></p>
<p><span class="math display">\[
(W/L)_{8}=(W/L)_{9},(W/L)_{10}=(W/L)_{11}
\]</span></p>
<p><span class="math display">\[
(W/L)_{12}=4(W/L)_{13}
\]</span></p>
<p><span class="math display">\[
(k_1+k_2+2)I_{DS8}\leq 400\mu A
\]</span></p>
<h2 id="单管匹配">单管匹配</h2>
<p>当晶体管工作在饱和区时，实际上的设计参数只有 <span class="math inline">\((W/L),I_{DS},V_{DSAT}\)</span>，确定了这三个变量中的两个就可以决定第三个量。在本设计中，确定晶体管的电流和过驱动电压就可以得到需要的宽长比。</p>
<figure>
<img src="https://pic.zhouyuqian.com/img/20210727183532.png" alt="cm"><figcaption aria-hidden="true">cm</figcaption>
</figure>
<p>如上图所示，首先单独搭建电流镜部分。</p>
<p>当 <span class="math inline">\(I_{DS} = 10\mu A\)</span> 时，扫描 W，得到在不同 <span class="math inline">\(W/L\)</span> 下 <span class="math inline">\(V_{DSAT}\)</span> 的值。</p>
<p><strong>PMOS</strong></p>
<table>
<thead>
<tr class="header">
<th><span class="math inline">\(V_{DSAT}\)</span></th>
<th><span class="math inline">\(W/L\)</span></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>100m</td>
<td>35</td>
</tr>
<tr class="even">
<td>125m</td>
<td>20</td>
</tr>
<tr class="odd">
<td>150m</td>
<td>13</td>
</tr>
<tr class="even">
<td>175m</td>
<td>9</td>
</tr>
<tr class="odd">
<td>200m</td>
<td>7</td>
</tr>
<tr class="even">
<td>250m</td>
<td>4</td>
</tr>
<tr class="odd">
<td>300m</td>
<td>3</td>
</tr>
</tbody>
</table>
<p><strong>NMOS</strong></p>
<table>
<thead>
<tr class="header">
<th><span class="math inline">\(V_{DSAT}\)</span></th>
<th><span class="math inline">\(W/L\)</span></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>100m</td>
<td>8.9</td>
</tr>
<tr class="even">
<td>125m</td>
<td>4.8</td>
</tr>
<tr class="odd">
<td>150m</td>
<td>3</td>
</tr>
<tr class="even">
<td>175m</td>
<td>2.1</td>
</tr>
<tr class="odd">
<td>250m</td>
<td>1</td>
</tr>
<tr class="even">
<td>200m</td>
<td>1.6</td>
</tr>
</tbody>
</table>
<p>可以看到在相同的过驱动电压下，<span class="math inline">\((W/L)_p/(W/L)_n \approx 13/3\)</span>，为了保证良好的匹配性，所有位于饱和区的晶体管取相同的过驱动电压。</p>
<h2 id="设计步骤">设计步骤</h2>
<ol type="1">
<li><p>先取 <span class="math inline">\(C_c\)</span> 为 1pF；</p></li>
<li><p>取 <span class="math inline">\(V_{DSAT} = 150mV\)</span>；</p></li>
<li><p>取 <span class="math inline">\(I_{DS8} = 10 \mu A, I_{DS5}=120\mu A, I_{DS7} = 240\mu A\)</span>；</p></li>
<li><p>确定晶体管参数，过驱动电压定为 150mV：</p>
<table>
<thead>
<tr class="header">
<th>晶体管</th>
<th>倍数</th>
<th>W/L</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>M13</td>
<td>1n</td>
<td>3/1</td>
</tr>
<tr class="even">
<td>M12</td>
<td>4n</td>
<td>12/1</td>
</tr>
<tr class="odd">
<td>M10, M11</td>
<td>1n</td>
<td>3/11</td>
</tr>
<tr class="even">
<td>M8, M9</td>
<td>1p</td>
<td>13/1</td>
</tr>
<tr class="odd">
<td>M5</td>
<td>12p</td>
<td>156/1</td>
</tr>
<tr class="even">
<td>M1, M2</td>
<td>6p</td>
<td>78/1</td>
</tr>
<tr class="odd">
<td>M3, M4</td>
<td>6n</td>
<td>18/1</td>
</tr>
<tr class="even">
<td>M7</td>
<td>24p</td>
<td>312/1</td>
</tr>
<tr class="odd">
<td>M6</td>
<td>24n</td>
<td>72/1</td>
</tr>
<tr class="even">
<td>M14</td>
<td></td>
<td>10/1</td>
</tr>
</tbody>
</table></li>
</ol>
<figure>
<img src="https://pic.zhouyuqian.com/img/20210727183533.png" alt="OTA"><figcaption aria-hidden="true">OTA</figcaption>
</figure>
<h1 id="cadence-仿真">cadence 仿真</h1>
<h2 id="ac">AC</h2>
<figure>
<img src="https://pic.zhouyuqian.com/img/20210727183534.svg" alt="ac"><figcaption aria-hidden="true">ac</figcaption>
</figure>
<p>通过 AC 仿真可以得到 GBW 大约为 113MHz，在单位带宽增益处，相移达到了 127°，表明相位裕度为 53°，不能满足大于 60° 的要求，因此调整米勒电容 Cc 为 1.5pF，再次仿真，得到GBW 为113MHz，相位裕度为 63°，满足要求。</p>
<figure>
<img src="https://pic.zhouyuqian.com/img/20210727183535.svg" alt="ac1.5p"><figcaption aria-hidden="true">ac1.5p</figcaption>
</figure>
<h2 id="噪声仿真">噪声仿真</h2>
<p><img src="https://pic.zhouyuqian.com/img/20210727183604.png" alt="截屏2021-03-20 下午4.47.15" style="zoom:33%;"></p>
<p>接下来仿真噪声，依照上图设置参数。其中，最下面 Positive Output Node 一栏，可点击 Select 按钮后，在电路图中选择输出节点；而 Negative Output Node 一栏则选地，若不选，则也会默认为地；Input Voltage Source 选择<strong>输入差分信号的电压源</strong>。结果选择等效输入噪声，如下：</p>
<p><img src="https://pic.zhouyuqian.com/img/20210727183623.svg" alt="noise"></p>
<h2 id="其他指标仿真">其他指标仿真</h2>
<figure>
<img src="https://pic.zhouyuqian.com/img/20210727183536.png" alt="tb_all"><figcaption aria-hidden="true">tb_all</figcaption>
</figure>
<p>SR、CMRR 和 nPSRR，仿真如上图所示。</p>
<h3 id="sr-压摆率">SR (压摆率)</h3>
<p><img src="https://pic.zhouyuqian.com/img/20210727183635.png" alt="截屏2021-03-20 下午6.43.01" style="zoom:33%;"></p>
<p>其中 SR 仿真中信号源使用 <code>vpulse</code>，设置如上图所示。仿真时使用 <code>tran</code> 仿真，仿真后输出波形。</p>
<figure>
<img src="https://pic.zhouyuqian.com/img/20210727183537.png" alt="截屏2021-03-20 下午6.37.13"><figcaption aria-hidden="true">截屏2021-03-20 下午6.37.13</figcaption>
</figure>
<p>由于 SR 是电压相对于时间的导数，可以使用计算器来得到一个更直观的 SR。 设置如上图所示。SR 的结果如下图所示。</p>
<figure>
<img src="https://pic.zhouyuqian.com/img/20210727183538.svg" alt="SR"><figcaption aria-hidden="true">SR</figcaption>
</figure>
<h3 id="cmrr-共模抑制比">CMRR (共模抑制比)</h3>
<p>定义为差模增益与差模输入为0时共模到输出的增益比值。物理意义即衡量共模输入转换到等效差模输入的大小。</p>
<p>使用两个信号源分别加载于被测放大器的同相、反相输入端，由于放大器的差模增益远远大于共模增益，因此 CMRR 可以近似为 <span class="math inline">\(V_{CM}/V_{out}\)</span>。 <span class="math display">\[
V_{out} = A_{DM}(V_P-V_N) + A_{CM}
\]</span> <span class="math display">\[
V_P= V_{CM},\ \ V_N = V_{out} + V_{CM}, \ \ V_{CM} = 1
\]</span> <span class="math display">\[
\frac{V_{out}}{V_{CM}} = \frac{A_{CM}}{A_{DM}+1}\approx\frac{A_{CM}}{A_{DM}}=\frac{1}{CMRR}
\]</span></p>
<figure>
<img src="https://pic.zhouyuqian.com/img/20210727183539.svg" alt="CMRR"><figcaption aria-hidden="true">CMRR</figcaption>
</figure>
<h3 id="电源抑制比-psrr">电源抑制比 (PSRR)</h3>
<p>在负电源上叠加 1V 交流电压，输出端接到负输入端，正输入端接 VDD/2 的直流电源。</p>
<figure>
<img src="https://pic.zhouyuqian.com/img/20210727183540.svg" alt="nPSRR"><figcaption aria-hidden="true">nPSRR</figcaption>
</figure>
<h2 id="仿真结果">仿真结果</h2>
<table>
<thead>
<tr class="header">
<th>设计指标</th>
<th>带宽最大化</th>
<th>仿真结果</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>静态功耗</td>
<td><span class="math inline">\(\leq 2mW\)</span></td>
<td><span class="math inline">\(377\mu A\times 1.8V\)</span></td>
</tr>
<tr class="even">
<td>开环直流增益</td>
<td><span class="math inline">\(\geq 80dB\)</span></td>
<td>83 dB</td>
</tr>
<tr class="odd">
<td>单位增益带宽</td>
<td>Maximize</td>
<td>113 MHz</td>
</tr>
<tr class="even">
<td>相位裕度</td>
<td><span class="math inline">\(\geq 60 degree\)</span></td>
<td>63 degree</td>
</tr>
<tr class="odd">
<td>转换速率</td>
<td><span class="math inline">\(\geq 30V/\mu s\)</span></td>
<td><span class="math inline">\(40 V/\mu s\)</span></td>
</tr>
<tr class="even">
<td>共模抑制比</td>
<td><span class="math inline">\(\geq 60dB\)</span></td>
<td>69 dB</td>
</tr>
<tr class="odd">
<td>负电源抑制比</td>
<td><span class="math inline">\(\geq 80dB\)</span></td>
<td>89 dB</td>
</tr>
<tr class="even">
<td>等效输入噪声</td>
<td><span class="math inline">\(\leq 300nV/Hz@1KHz\)</span></td>
<td><span class="math inline">\(91.6nV/Hz@1KHz\)</span></td>
</tr>
</tbody>
</table>
<h1 id="reference">Reference</h1>
<p>[1] <a target="_blank" rel="noopener" href="http://bbs.eetop.cn/thread-279034-1-1.html">复旦大学模拟电路二级运放实例</a></p>
<p>[2] <a target="_blank" rel="noopener" href="http://blog.eetop.cn/blog-1770479-6946353.html">两级运算放大器设计笔记</a></p>
<p>[3] <a target="_blank" rel="noopener" href="https://www.bilibili.com/read/cv5808960/">cadence中运放性能参数仿真（一）</a></p>
<p>[4] <a target="_blank" rel="noopener" href="https://wx.seu.edu.cn/_upload/article/files/ed/b2/3c80f23a41e79abc3114a231ed52/9c379d42-4ec3-4605-9ea7-a55b922bd984.pdf">运放仿真方法整理</a></p>

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        <div style="text-align:center;color: #ccc;font-size:14px;">-------The end of this article&nbsp;<i class="fa fa-paw"></i>&nbsp;Thank you for your reading-------</div>
    
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    <strong>本文链接：</strong>
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    <strong>版权声明： </strong>本博客所有文章除特别声明外，均采用 <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/zh" rel="noopener" target="_blank"><i class="fa fa-fw fa-creative-commons"></i>BY-NC-SA</a> 许可协议。转载请注明出处！
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